Having learnt about the dynamics of a single Möbius map in the last chapter, we now embark on the topic which will occupy almost all the rest of the book: what patterns are simultaneously symmetrical under two Möbius maps? This turns out to be a very fruitful question, because two transformations can interact in several very different ways. They will try their best to dance together – but they do not always succeed. Like two friends making music, they may perform in simple harmony, they may be elaborately contrapuntal, or the result may be total dissonance. This chapter studies one of the simplest possible arrangements. As we go further, the full range of complexity from relative order to total chaos will gradually unfold.

In Chapter 1, we studied the collection of all possible transformations obtained by composing (that is, iterating) two initial transformations and their inverses in any order whatsoever. At that point, we were talking about Euclidean symmetries which preserved a pattern of tiles on a bathroom floor, but there is no reason why what we said there should not equally be applied to Moöbius transformations which, as we have seen, can be thought of as symmetries of patterns on the Riemann sphere.